Find-the-limit-limit-1-mt-2-t-t-

Question Number 5593 by sanusihammed last updated on 21/May/16

F i n d t h e l i m i t

l i m i t {[1 + m t]}^{\frac{2}{t}}

t \to \infty

Answered by FilupSmith last updated on 21/May/16

L = \lim_{t \to \infty} {(1 + m t)}^{2 / t}

= \lim_{t \to \infty} e^{\ln ({(1 + m t)}^{2 / t})}

= \lim_{t \to \infty} e^{\frac{2}{t} \ln (1 + m t)}

e^{x} = \exp (x) c h a n g e o f n o t a t i o n f o r

e a s i e r w r i t i n g

= \lim_{t \to \infty} \exp (\frac{2 \ln (1 + m t)}{t})

= \exp (\lim_{t \to \infty} \frac{2 \ln (1 + m t)}{t})

use L^{'} {H o p i t a l}^{'} s L a w

L = \exp (\lim_{t \to \infty} \frac{2 \times k}{1})

k = \frac{d}{d t} (\ln (1 + m t))

l e t u = 1 + m t \Rightarrow d u = m d t

k = \frac{d}{d t} m \ln (u)

k = \frac{m}{1 + m t}

∴ L = \exp (\lim_{t \to \infty} \frac{2 \frac{m}{1 + m t}}{1})

L = \exp (\lim_{t \to \infty} 2 \frac{m}{1 + m t})

L = \exp (2 \times 0)

L = 1

∴ \lim_{t \to \infty} {(1 + m t)}^{2 / t} = 1

Commented by FilupSmith last updated on 22/May/16

Not that I can work out

Commented by FilupSmith last updated on 22/May/16

No problem

Commented by sanusihammed last updated on 21/May/16

T h a n k s s o m u c h . G o d b l e s s y o u

Commented by Rasheed Soomro last updated on 22/May/16

Could the problem be solved without

using calculus ?